# Properties

 Label 1380.2.a.e.1.1 Level $1380$ Weight $2$ Character 1380.1 Self dual yes Analytic conductor $11.019$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1,1380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$11.0193554789$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1380.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.00000 q^{13} +1.00000 q^{15} -7.00000 q^{17} -6.00000 q^{19} -3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} +9.00000 q^{31} -2.00000 q^{33} -3.00000 q^{35} -7.00000 q^{37} -2.00000 q^{39} +5.00000 q^{41} +1.00000 q^{45} +8.00000 q^{47} +2.00000 q^{49} -7.00000 q^{51} -11.0000 q^{53} -2.00000 q^{55} -6.00000 q^{57} +9.00000 q^{59} -3.00000 q^{63} -2.00000 q^{65} -3.00000 q^{67} +1.00000 q^{69} +3.00000 q^{71} -6.00000 q^{73} +1.00000 q^{75} +6.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +5.00000 q^{83} -7.00000 q^{85} -9.00000 q^{87} +6.00000 q^{91} +9.00000 q^{93} -6.00000 q^{95} -10.0000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 9.00000 1.61645 0.808224 0.588875i $$-0.200429\pi$$
0.808224 + 0.588875i $$0.200429\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −7.00000 −0.980196
$$52$$ 0 0
$$53$$ −11.0000 −1.51097 −0.755483 0.655168i $$-0.772598\pi$$
−0.755483 + 0.655168i $$0.772598\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ −6.00000 −0.794719
$$58$$ 0 0
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ −3.00000 −0.377964
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 3.00000 0.356034 0.178017 0.984027i $$-0.443032\pi$$
0.178017 + 0.984027i $$0.443032\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 5.00000 0.548821 0.274411 0.961613i $$-0.411517\pi$$
0.274411 + 0.961613i $$0.411517\pi$$
$$84$$ 0 0
$$85$$ −7.00000 −0.759257
$$86$$ 0 0
$$87$$ −9.00000 −0.964901
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ 0 0
$$93$$ 9.00000 0.933257
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ 5.00000 0.483368 0.241684 0.970355i $$-0.422300\pi$$
0.241684 + 0.970355i $$0.422300\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −7.00000 −0.664411
$$112$$ 0 0
$$113$$ 21.0000 1.97551 0.987757 0.156001i $$-0.0498603\pi$$
0.987757 + 0.156001i $$0.0498603\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 21.0000 1.92507
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 5.00000 0.450835
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 18.0000 1.56080
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ −9.00000 −0.747409
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 0 0
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ −7.00000 −0.565916
$$154$$ 0 0
$$155$$ 9.00000 0.722897
$$156$$ 0 0
$$157$$ −5.00000 −0.399043 −0.199522 0.979893i $$-0.563939\pi$$
−0.199522 + 0.979893i $$0.563939\pi$$
$$158$$ 0 0
$$159$$ −11.0000 −0.872357
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ −14.0000 −1.09656 −0.548282 0.836293i $$-0.684718\pi$$
−0.548282 + 0.836293i $$0.684718\pi$$
$$164$$ 0 0
$$165$$ −2.00000 −0.155700
$$166$$ 0 0
$$167$$ −4.00000 −0.309529 −0.154765 0.987951i $$-0.549462\pi$$
−0.154765 + 0.987951i $$0.549462\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −6.00000 −0.458831
$$172$$ 0 0
$$173$$ 4.00000 0.304114 0.152057 0.988372i $$-0.451410\pi$$
0.152057 + 0.988372i $$0.451410\pi$$
$$174$$ 0 0
$$175$$ −3.00000 −0.226779
$$176$$ 0 0
$$177$$ 9.00000 0.676481
$$178$$ 0 0
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −7.00000 −0.514650
$$186$$ 0 0
$$187$$ 14.0000 1.02378
$$188$$ 0 0
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ −24.0000 −1.72756 −0.863779 0.503871i $$-0.831909\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 0 0
$$195$$ −2.00000 −0.143223
$$196$$ 0 0
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ 0 0
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ −3.00000 −0.211604
$$202$$ 0 0
$$203$$ 27.0000 1.89503
$$204$$ 0 0
$$205$$ 5.00000 0.349215
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −23.0000 −1.58339 −0.791693 0.610920i $$-0.790800\pi$$
−0.791693 + 0.610920i $$0.790800\pi$$
$$212$$ 0 0
$$213$$ 3.00000 0.205557
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −27.0000 −1.83288
$$218$$ 0 0
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ 14.0000 0.941742
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 16.0000 1.04819 0.524097 0.851658i $$-0.324403\pi$$
0.524097 + 0.851658i $$0.324403\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 21.0000 1.35838 0.679189 0.733964i $$-0.262332\pi$$
0.679189 + 0.733964i $$0.262332\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ 12.0000 0.763542
$$248$$ 0 0
$$249$$ 5.00000 0.316862
$$250$$ 0 0
$$251$$ −30.0000 −1.89358 −0.946792 0.321847i $$-0.895696\pi$$
−0.946792 + 0.321847i $$0.895696\pi$$
$$252$$ 0 0
$$253$$ −2.00000 −0.125739
$$254$$ 0 0
$$255$$ −7.00000 −0.438357
$$256$$ 0 0
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ 0 0
$$259$$ 21.0000 1.30488
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 0 0
$$263$$ 11.0000 0.678289 0.339145 0.940734i $$-0.389862\pi$$
0.339145 + 0.940734i $$0.389862\pi$$
$$264$$ 0 0
$$265$$ −11.0000 −0.675725
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 0 0
$$271$$ 31.0000 1.88312 0.941558 0.336851i $$-0.109362\pi$$
0.941558 + 0.336851i $$0.109362\pi$$
$$272$$ 0 0
$$273$$ 6.00000 0.363137
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 0 0
$$279$$ 9.00000 0.538816
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −5.00000 −0.297219 −0.148610 0.988896i $$-0.547480\pi$$
−0.148610 + 0.988896i $$0.547480\pi$$
$$284$$ 0 0
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ −15.0000 −0.885422
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 0 0
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ 0 0
$$295$$ 9.00000 0.524000
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 3.00000 0.172345
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −30.0000 −1.71219 −0.856095 0.516818i $$-0.827116\pi$$
−0.856095 + 0.516818i $$0.827116\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −29.0000 −1.63918 −0.819588 0.572953i $$-0.805798\pi$$
−0.819588 + 0.572953i $$0.805798\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ 5.00000 0.279073
$$322$$ 0 0
$$323$$ 42.0000 2.33694
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 10.0000 0.553001
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 21.0000 1.15426 0.577132 0.816651i $$-0.304172\pi$$
0.577132 + 0.816651i $$0.304172\pi$$
$$332$$ 0 0
$$333$$ −7.00000 −0.383598
$$334$$ 0 0
$$335$$ −3.00000 −0.163908
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 0 0
$$339$$ 21.0000 1.14056
$$340$$ 0 0
$$341$$ −18.0000 −0.974755
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ 1.00000 0.0538382
$$346$$ 0 0
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ 31.0000 1.65939 0.829696 0.558216i $$-0.188514\pi$$
0.829696 + 0.558216i $$0.188514\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ −28.0000 −1.49029 −0.745145 0.666903i $$-0.767620\pi$$
−0.745145 + 0.666903i $$0.767620\pi$$
$$354$$ 0 0
$$355$$ 3.00000 0.159223
$$356$$ 0 0
$$357$$ 21.0000 1.11144
$$358$$ 0 0
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 7.00000 0.365397 0.182699 0.983169i $$-0.441517\pi$$
0.182699 + 0.983169i $$0.441517\pi$$
$$368$$ 0 0
$$369$$ 5.00000 0.260290
$$370$$ 0 0
$$371$$ 33.0000 1.71327
$$372$$ 0 0
$$373$$ 34.0000 1.76045 0.880227 0.474554i $$-0.157390\pi$$
0.880227 + 0.474554i $$0.157390\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 18.0000 0.927047
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ 27.0000 1.37964 0.689818 0.723983i $$-0.257691\pi$$
0.689818 + 0.723983i $$0.257691\pi$$
$$384$$ 0 0
$$385$$ 6.00000 0.305788
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −7.00000 −0.354005
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ −28.0000 −1.40528 −0.702640 0.711546i $$-0.747995\pi$$
−0.702640 + 0.711546i $$0.747995\pi$$
$$398$$ 0 0
$$399$$ 18.0000 0.901127
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ −18.0000 −0.896644
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 14.0000 0.693954
$$408$$ 0 0
$$409$$ −35.0000 −1.73064 −0.865319 0.501221i $$-0.832884\pi$$
−0.865319 + 0.501221i $$0.832884\pi$$
$$410$$ 0 0
$$411$$ −14.0000 −0.690569
$$412$$ 0 0
$$413$$ −27.0000 −1.32858
$$414$$ 0 0
$$415$$ 5.00000 0.245440
$$416$$ 0 0
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ 0 0
$$423$$ 8.00000 0.388973
$$424$$ 0 0
$$425$$ −7.00000 −0.339550
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ −4.00000 −0.192673 −0.0963366 0.995349i $$-0.530713\pi$$
−0.0963366 + 0.995349i $$0.530713\pi$$
$$432$$ 0 0
$$433$$ 1.00000 0.0480569 0.0240285 0.999711i $$-0.492351\pi$$
0.0240285 + 0.999711i $$0.492351\pi$$
$$434$$ 0 0
$$435$$ −9.00000 −0.431517
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −4.00000 −0.189194
$$448$$ 0 0
$$449$$ −17.0000 −0.802280 −0.401140 0.916017i $$-0.631386\pi$$
−0.401140 + 0.916017i $$0.631386\pi$$
$$450$$ 0 0
$$451$$ −10.0000 −0.470882
$$452$$ 0 0
$$453$$ −4.00000 −0.187936
$$454$$ 0 0
$$455$$ 6.00000 0.281284
$$456$$ 0 0
$$457$$ 7.00000 0.327446 0.163723 0.986506i $$-0.447650\pi$$
0.163723 + 0.986506i $$0.447650\pi$$
$$458$$ 0 0
$$459$$ −7.00000 −0.326732
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ −10.0000 −0.464739 −0.232370 0.972628i $$-0.574648\pi$$
−0.232370 + 0.972628i $$0.574648\pi$$
$$464$$ 0 0
$$465$$ 9.00000 0.417365
$$466$$ 0 0
$$467$$ 21.0000 0.971764 0.485882 0.874024i $$-0.338498\pi$$
0.485882 + 0.874024i $$0.338498\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ −5.00000 −0.230388
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 0 0
$$477$$ −11.0000 −0.503655
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 14.0000 0.638345
$$482$$ 0 0
$$483$$ −3.00000 −0.136505
$$484$$ 0 0
$$485$$ −10.0000 −0.454077
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ −14.0000 −0.633102
$$490$$ 0 0
$$491$$ −5.00000 −0.225647 −0.112823 0.993615i $$-0.535989\pi$$
−0.112823 + 0.993615i $$0.535989\pi$$
$$492$$ 0 0
$$493$$ 63.0000 2.83738
$$494$$ 0 0
$$495$$ −2.00000 −0.0898933
$$496$$ 0 0
$$497$$ −9.00000 −0.403705
$$498$$ 0 0
$$499$$ 15.0000 0.671492 0.335746 0.941953i $$-0.391012\pi$$
0.335746 + 0.941953i $$0.391012\pi$$
$$500$$ 0 0
$$501$$ −4.00000 −0.178707
$$502$$ 0 0
$$503$$ −33.0000 −1.47140 −0.735699 0.677309i $$-0.763146\pi$$
−0.735699 + 0.677309i $$0.763146\pi$$
$$504$$ 0 0
$$505$$ 3.00000 0.133498
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 18.0000 0.796273
$$512$$ 0 0
$$513$$ −6.00000 −0.264906
$$514$$ 0 0
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ −16.0000 −0.703679
$$518$$ 0 0
$$519$$ 4.00000 0.175581
$$520$$ 0 0
$$521$$ 38.0000 1.66481 0.832405 0.554168i $$-0.186963\pi$$
0.832405 + 0.554168i $$0.186963\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 0 0
$$525$$ −3.00000 −0.130931
$$526$$ 0 0
$$527$$ −63.0000 −2.74432
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 9.00000 0.390567
$$532$$ 0 0
$$533$$ −10.0000 −0.433148
$$534$$ 0 0
$$535$$ 5.00000 0.216169
$$536$$ 0 0
$$537$$ 16.0000 0.690451
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 0 0
$$543$$ 12.0000 0.514969
$$544$$ 0 0
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ −24.0000 −1.02617 −0.513083 0.858339i $$-0.671497\pi$$
−0.513083 + 0.858339i $$0.671497\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 54.0000 2.30048
$$552$$ 0 0
$$553$$ 24.0000 1.02058
$$554$$ 0 0
$$555$$ −7.00000 −0.297133
$$556$$ 0 0
$$557$$ −11.0000 −0.466085 −0.233042 0.972467i $$-0.574868\pi$$
−0.233042 + 0.972467i $$0.574868\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 14.0000 0.591080
$$562$$ 0 0
$$563$$ −5.00000 −0.210725 −0.105362 0.994434i $$-0.533600\pi$$
−0.105362 + 0.994434i $$0.533600\pi$$
$$564$$ 0 0
$$565$$ 21.0000 0.883477
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 8.00000 0.334790 0.167395 0.985890i $$-0.446465\pi$$
0.167395 + 0.985890i $$0.446465\pi$$
$$572$$ 0 0
$$573$$ 4.00000 0.167102
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −32.0000 −1.33218 −0.666089 0.745873i $$-0.732033\pi$$
−0.666089 + 0.745873i $$0.732033\pi$$
$$578$$ 0 0
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ −15.0000 −0.622305
$$582$$ 0 0
$$583$$ 22.0000 0.911147
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ −2.00000 −0.0825488 −0.0412744 0.999148i $$-0.513142\pi$$
−0.0412744 + 0.999148i $$0.513142\pi$$
$$588$$ 0 0
$$589$$ −54.0000 −2.22503
$$590$$ 0 0
$$591$$ 14.0000 0.575883
$$592$$ 0 0
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ 21.0000 0.860916
$$596$$ 0 0
$$597$$ 14.0000 0.572982
$$598$$ 0 0
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ 0 0
$$603$$ −3.00000 −0.122169
$$604$$ 0 0
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ 27.0000 1.09410
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ 0 0
$$613$$ −14.0000 −0.565455 −0.282727 0.959200i $$-0.591239\pi$$
−0.282727 + 0.959200i $$0.591239\pi$$
$$614$$ 0 0
$$615$$ 5.00000 0.201619
$$616$$ 0 0
$$617$$ −9.00000 −0.362326 −0.181163 0.983453i $$-0.557986\pi$$
−0.181163 + 0.983453i $$0.557986\pi$$
$$618$$ 0 0
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 12.0000 0.479234
$$628$$ 0 0
$$629$$ 49.0000 1.95376
$$630$$ 0 0
$$631$$ −44.0000 −1.75161 −0.875806 0.482663i $$-0.839670\pi$$
−0.875806 + 0.482663i $$0.839670\pi$$
$$632$$ 0 0
$$633$$ −23.0000 −0.914168
$$634$$ 0 0
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ −4.00000 −0.158486
$$638$$ 0 0
$$639$$ 3.00000 0.118678
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ −23.0000 −0.907031 −0.453516 0.891248i $$-0.649830\pi$$
−0.453516 + 0.891248i $$0.649830\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 10.0000 0.393141 0.196570 0.980490i $$-0.437020\pi$$
0.196570 + 0.980490i $$0.437020\pi$$
$$648$$ 0 0
$$649$$ −18.0000 −0.706562
$$650$$ 0 0
$$651$$ −27.0000 −1.05821
$$652$$ 0 0
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ 0 0
$$663$$ 14.0000 0.543715
$$664$$ 0 0
$$665$$ 18.0000 0.698010
$$666$$ 0 0
$$667$$ −9.00000 −0.348481
$$668$$ 0 0
$$669$$ −2.00000 −0.0773245
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 16.0000 0.616755 0.308377 0.951264i $$-0.400214\pi$$
0.308377 + 0.951264i $$0.400214\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −21.0000 −0.807096 −0.403548 0.914959i $$-0.632223\pi$$
−0.403548 + 0.914959i $$0.632223\pi$$
$$678$$ 0 0
$$679$$ 30.0000 1.15129
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ −14.0000 −0.534913
$$686$$ 0 0
$$687$$ −16.0000 −0.610438
$$688$$ 0 0
$$689$$ 22.0000 0.838133
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ 0 0
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ −5.00000 −0.189661
$$696$$ 0 0
$$697$$ −35.0000 −1.32572
$$698$$ 0 0
$$699$$ 16.0000 0.605176
$$700$$ 0 0
$$701$$ −20.0000 −0.755390 −0.377695 0.925930i $$-0.623283\pi$$
−0.377695 + 0.925930i $$0.623283\pi$$
$$702$$ 0 0
$$703$$ 42.0000 1.58406
$$704$$ 0 0
$$705$$ 8.00000 0.301297
$$706$$ 0 0
$$707$$ −9.00000 −0.338480
$$708$$ 0 0
$$709$$ −32.0000 −1.20179 −0.600893 0.799330i $$-0.705188\pi$$
−0.600893 + 0.799330i $$0.705188\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 9.00000 0.337053
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ 0 0
$$717$$ 21.0000 0.784259
$$718$$ 0 0
$$719$$ −43.0000 −1.60363 −0.801815 0.597573i $$-0.796132\pi$$
−0.801815 + 0.597573i $$0.796132\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ 0 0
$$723$$ 26.0000 0.966950
$$724$$ 0 0
$$725$$ −9.00000 −0.334252
$$726$$ 0 0
$$727$$ 23.0000 0.853023 0.426511 0.904482i $$-0.359742\pi$$
0.426511 + 0.904482i $$0.359742\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −47.0000 −1.73598 −0.867992 0.496578i $$-0.834590\pi$$
−0.867992 + 0.496578i $$0.834590\pi$$
$$734$$ 0 0
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ 6.00000 0.221013
$$738$$ 0 0
$$739$$ −23.0000 −0.846069 −0.423034 0.906114i $$-0.639035\pi$$
−0.423034 + 0.906114i $$0.639035\pi$$
$$740$$ 0 0
$$741$$ 12.0000 0.440831
$$742$$ 0 0
$$743$$ −32.0000 −1.17397 −0.586983 0.809599i $$-0.699684\pi$$
−0.586983 + 0.809599i $$0.699684\pi$$
$$744$$ 0 0
$$745$$ −4.00000 −0.146549
$$746$$ 0 0
$$747$$ 5.00000 0.182940
$$748$$ 0 0
$$749$$ −15.0000 −0.548088
$$750$$ 0 0
$$751$$ 10.0000 0.364905 0.182453 0.983215i $$-0.441596\pi$$
0.182453 + 0.983215i $$0.441596\pi$$
$$752$$ 0 0
$$753$$ −30.0000 −1.09326
$$754$$ 0 0
$$755$$ −4.00000 −0.145575
$$756$$ 0 0
$$757$$ −1.00000 −0.0363456 −0.0181728 0.999835i $$-0.505785\pi$$
−0.0181728 + 0.999835i $$0.505785\pi$$
$$758$$ 0 0
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ −39.0000 −1.41375 −0.706874 0.707339i $$-0.749895\pi$$
−0.706874 + 0.707339i $$0.749895\pi$$
$$762$$ 0 0
$$763$$ −30.0000 −1.08607
$$764$$ 0 0
$$765$$ −7.00000 −0.253086
$$766$$ 0 0
$$767$$ −18.0000 −0.649942
$$768$$ 0 0
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ −8.00000 −0.288113
$$772$$ 0 0
$$773$$ −54.0000 −1.94225 −0.971123 0.238581i $$-0.923318\pi$$
−0.971123 + 0.238581i $$0.923318\pi$$
$$774$$ 0 0
$$775$$ 9.00000 0.323290
$$776$$ 0 0
$$777$$ 21.0000 0.753371
$$778$$ 0 0
$$779$$ −30.0000 −1.07486
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ 0 0
$$783$$ −9.00000 −0.321634
$$784$$ 0 0
$$785$$ −5.00000 −0.178458
$$786$$ 0 0
$$787$$ −31.0000 −1.10503 −0.552515 0.833503i $$-0.686332\pi$$
−0.552515 + 0.833503i $$0.686332\pi$$
$$788$$ 0 0
$$789$$ 11.0000 0.391610
$$790$$ 0 0
$$791$$ −63.0000 −2.24002
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −11.0000 −0.390130
$$796$$ 0 0
$$797$$ 11.0000 0.389640 0.194820 0.980839i $$-0.437588\pi$$
0.194820 + 0.980839i $$0.437588\pi$$
$$798$$ 0 0
$$799$$ −56.0000 −1.98114
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 12.0000 0.423471
$$804$$ 0 0
$$805$$ −3.00000 −0.105736
$$806$$ 0 0
$$807$$ −3.00000 −0.105605
$$808$$ 0 0
$$809$$ 3.00000 0.105474 0.0527372 0.998608i $$-0.483205\pi$$
0.0527372 + 0.998608i $$0.483205\pi$$
$$810$$ 0 0
$$811$$ −7.00000 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$812$$ 0 0
$$813$$ 31.0000 1.08722
$$814$$ 0 0
$$815$$ −14.0000 −0.490399
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 6.00000 0.209657
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 52.0000 1.81261 0.906303 0.422628i $$-0.138892\pi$$
0.906303 + 0.422628i $$0.138892\pi$$
$$824$$ 0 0
$$825$$ −2.00000 −0.0696311
$$826$$ 0 0
$$827$$ 31.0000 1.07798 0.538988 0.842314i $$-0.318807\pi$$
0.538988 + 0.842314i $$0.318807\pi$$
$$828$$ 0 0
$$829$$ 49.0000 1.70184 0.850920 0.525295i $$-0.176045\pi$$
0.850920 + 0.525295i $$0.176045\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ 0 0
$$833$$ −14.0000 −0.485071
$$834$$ 0 0
$$835$$ −4.00000 −0.138426
$$836$$ 0 0
$$837$$ 9.00000 0.311086
$$838$$ 0 0
$$839$$ 2.00000 0.0690477 0.0345238 0.999404i $$-0.489009\pi$$
0.0345238 + 0.999404i $$0.489009\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 10.0000 0.344418
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 21.0000 0.721569
$$848$$ 0 0
$$849$$ −5.00000 −0.171600
$$850$$ 0 0
$$851$$ −7.00000 −0.239957
$$852$$ 0 0
$$853$$ −30.0000 −1.02718 −0.513590 0.858036i $$-0.671685\pi$$
−0.513590 + 0.858036i $$0.671685\pi$$
$$854$$ 0 0
$$855$$ −6.00000 −0.205196
$$856$$ 0 0
$$857$$ 54.0000 1.84460 0.922302 0.386469i $$-0.126305\pi$$
0.922302 + 0.386469i $$0.126305\pi$$
$$858$$ 0 0
$$859$$ −43.0000 −1.46714 −0.733571 0.679613i $$-0.762148\pi$$
−0.733571 + 0.679613i $$0.762148\pi$$
$$860$$ 0 0
$$861$$ −15.0000 −0.511199
$$862$$ 0 0
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ 0 0
$$865$$ 4.00000 0.136004
$$866$$ 0 0
$$867$$ 32.0000 1.08678
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 6.00000 0.203302
$$872$$ 0 0
$$873$$ −10.0000 −0.338449
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ 0 0
$$879$$ −9.00000 −0.303562
$$880$$ 0 0
$$881$$ −16.0000 −0.539054 −0.269527 0.962993i $$-0.586867\pi$$
−0.269527 + 0.962993i $$0.586867\pi$$
$$882$$ 0 0
$$883$$ 12.0000 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$884$$ 0 0
$$885$$ 9.00000 0.302532
$$886$$ 0 0
$$887$$ 38.0000 1.27592 0.637958 0.770072i $$-0.279780\pi$$
0.637958 + 0.770072i $$0.279780\pi$$
$$888$$ 0 0
$$889$$ 48.0000 1.60987
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ −48.0000 −1.60626
$$894$$ 0 0
$$895$$ 16.0000 0.534821
$$896$$ 0 0
$$897$$ −2.00000 −0.0667781
$$898$$ 0 0
$$899$$ −81.0000 −2.70150
$$900$$ 0 0
$$901$$ 77.0000 2.56524
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 12.0000 0.398893
$$906$$ 0 0
$$907$$ 37.0000 1.22856 0.614282 0.789086i $$-0.289446\pi$$
0.614282 + 0.789086i $$0.289446\pi$$
$$908$$ 0 0
$$909$$ 3.00000 0.0995037
$$910$$ 0 0
$$911$$ −56.0000 −1.85536 −0.927681 0.373373i $$-0.878201\pi$$
−0.927681 + 0.373373i $$0.878201\pi$$
$$912$$ 0 0
$$913$$ −10.0000 −0.330952
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 36.0000 1.18882
$$918$$ 0 0
$$919$$ −12.0000 −0.395843 −0.197922 0.980218i $$-0.563419\pi$$
−0.197922 + 0.980218i $$0.563419\pi$$
$$920$$ 0 0
$$921$$ −30.0000 −0.988534
$$922$$ 0 0
$$923$$ −6.00000 −0.197492
$$924$$ 0 0
$$925$$ −7.00000 −0.230159
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ −57.0000 −1.87011 −0.935055 0.354504i $$-0.884650\pi$$
−0.935055 + 0.354504i $$0.884650\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 14.0000 0.457849
$$936$$ 0 0
$$937$$ 46.0000 1.50275 0.751377 0.659873i $$-0.229390\pi$$
0.751377 + 0.659873i $$0.229390\pi$$
$$938$$ 0 0
$$939$$ −29.0000 −0.946379
$$940$$ 0 0
$$941$$ −8.00000 −0.260793 −0.130396 0.991462i $$-0.541625\pi$$
−0.130396 + 0.991462i $$0.541625\pi$$
$$942$$ 0 0
$$943$$ 5.00000 0.162822
$$944$$ 0 0
$$945$$ −3.00000 −0.0975900
$$946$$ 0 0
$$947$$ −24.0000 −0.779895 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ 4.00000 0.129437
$$956$$ 0 0
$$957$$ 18.0000 0.581857
$$958$$ 0 0
$$959$$ 42.0000 1.35625
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ 0 0
$$963$$ 5.00000 0.161123
$$964$$ 0 0
$$965$$ −24.0000 −0.772587
$$966$$ 0 0
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ 0 0
$$969$$ 42.0000 1.34923
$$970$$ 0 0
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 0 0
$$973$$ 15.0000 0.480878
$$974$$ 0 0
$$975$$ −2.00000 −0.0640513
$$976$$ 0 0
$$977$$ −37.0000 −1.18373 −0.591867 0.806035i $$-0.701609\pi$$
−0.591867 + 0.806035i $$0.701609\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 0 0
$$983$$ 41.0000 1.30770 0.653848 0.756626i $$-0.273153\pi$$
0.653848 + 0.756626i $$0.273153\pi$$
$$984$$ 0 0
$$985$$ 14.0000 0.446077
$$986$$ 0 0
$$987$$ −24.0000 −0.763928
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −59.0000 −1.87420 −0.937098 0.349065i $$-0.886499\pi$$
−0.937098 + 0.349065i $$0.886499\pi$$
$$992$$ 0 0
$$993$$ 21.0000 0.666415
$$994$$ 0 0
$$995$$ 14.0000 0.443830
$$996$$ 0 0
$$997$$ 16.0000 0.506725 0.253363 0.967371i $$-0.418463\pi$$
0.253363 + 0.967371i $$0.418463\pi$$
$$998$$ 0 0
$$999$$ −7.00000 −0.221470
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1380.2.a.e.1.1 1
3.2 odd 2 4140.2.a.b.1.1 1
4.3 odd 2 5520.2.a.m.1.1 1
5.2 odd 4 6900.2.f.b.6349.1 2
5.3 odd 4 6900.2.f.b.6349.2 2
5.4 even 2 6900.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.e.1.1 1 1.1 even 1 trivial
4140.2.a.b.1.1 1 3.2 odd 2
5520.2.a.m.1.1 1 4.3 odd 2
6900.2.a.c.1.1 1 5.4 even 2
6900.2.f.b.6349.1 2 5.2 odd 4
6900.2.f.b.6349.2 2 5.3 odd 4